On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces

Sahami, Amir; Rostami, Mehdi; Shariati, Seyedeh Fatemeh; Babayi, Salman

doi:https://doi.org/10.1155/2022/4939971

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Abstract Introduction and Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 4939971 | https://doi.org/10.1155/2022/4939971

On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces

Amir Sahami,¹Mehdi Rostami,²Seyedeh Fatemeh Shariati,²and Salman Babayi³

Academic Editor: Ralf Meyer

Received25 Oct 2021

Revised08 Dec 2021

Accepted06 Jan 2022

Published29 Jan 2022

Abstract

In this paper, we study the notion of approximate biprojectivity and left -biprojectivity of some Banach algebras, where is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra implies that is compact. Moreover, we investigate left -biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra is left -biprojective if and only if is compact, where is a hypergroup. We also study the approximate biflatness and left -biflatness of hypergroup algebras in terms of amenability of their related hypergroups.

1. Introduction and Preliminaries

Hypergroups are a suitable generalization of classical locally compact groups. In classical setting, the convolution of two point mass measures is a point mass measure, while in hypergroup structure, the convolution of two point mass measures is a probability measure with compact support. The study of hypergroups was initiated in the 1970s by Dunkl [1], Jewwet [2], and Spector [3], each of them in various axioms. However, in this paper, we will base our work on Jewett’s axioms in [2].

Biprojectivity is an important homological notion that arises naturally in Helemskii’s works in the 1980s; interested readers can refer to his comprehensive book [4]. Biprojectivity of some well-known Banach algebras associated to locally compact groups, such as group algebras and measure algebras, is studied in [4, 5]. Biprojectivity of the hypergroup algebra is studied in [6]. As a generalization of this notion, Y. Zhang in [7] introduced the notion of approximate biprojectivity. Indeed, a Banach algebra is called approximately biprojective if there exists a net of continuous -bimodule morphism from into such that for every , where is the diagonal operator defined by . For recent works about this concept, refer to [8].

Throughout the paper, stands for the set of all nonzero multiplicative linear functionals on . Kaniuth et al. [9] introduced the notion of left -amenable Banach algebras as a generalization of the notion of amenable Banach algebras introduced by Johnson in [10]. A Banach algebra is called left -amenable if every derivation from into is inner, for every Banach -bimodule with the left module action for all and .

Hu et al. in [11] defined the notion of left -contractibility for Banach algebras. Following [12], a Banach algebra is called left -contractible, where , if there exists such that and , for every . For a locally compact group , it is shown that left -contractibility of (or ) is equivalent to compactness of (Theorem 6.1 in [15]).

Motivated by these considerations, the first author defined the homological notion of left -biprojectivity for Banach algebras (see, e.g., [13]). Here is the definition of his new notion. A Banach algebra is called left -biprojective, where , if there exists a bounded linear map such that

The right case can be similarly defined.

The first and the second authors in [13] explained the relation between left -contractibility and left -biprojectivity. They proved that left -contractibility implies left -biprojectivity, and the converse is valid if is commutative or has a left approximate identity.

We give some brief backgrounds on hypergroups and their related algebras and establish our notation; for details, see [14]. Let be a locally compact Hausdorff space and denote the set of all bounded complex Radon measures on , where the norm of each measure is the total variation . Also, we denote by the set of all nonempty compact subsets of and equipped this space with the Michael topology, that is, the topology generated by the setsfor open subsets and in .

The space is called a hypergroup if there is a convolution , an involution on , and an element (called the identity element) such that the following holds:(i) is a Banach algebra.(ii) is a probability measure with compact support.(iii)The map is continuous from into equipped with the weak topology.(iv)The map is continuous from into .(v)For each , .(vi)The mapping is a homeomorphism on of period 2 and and if and only if for each . Here, the measure is given by for all Borel subsets .

A hypergroup is called commutative if for all . In [2, 3], it is proved that every commutative or compact hypergroup has a unique (left) Haar measure. The existence and the uniqueness of left Haar measure on a general locally compact hypergroup were proved recently by Chapovsky in [15]. Throughout this paper, we assume that is a hypergroup with a unique Haar measure . That is,for every Borel measurable function on . Then, with the involution and the convolutionis a Banach -algebra, where for every . A nonzero bounded continuous function is called a character on if for every . The set of all characters on will be denoted by , i.e.,

For , define on via

It may be observed that , and note that there is at least one character on , namely, the augmentation character . Further, if is commutative, it is well-known that there is no other character on ; that is, . For more details, see Section 2.2 in [3].

In the present paper, we show that approximate biprojectivity of hypergroup algebra implies that is compact. After that, we study left -biprojectivity of general abstract Segal algebras with respect to the hypergroup algebra . As a result (with a mild condition), we conclude that the abstract Segal algebra is left -biprojective if and only if is compact. We also study approximate biflatness and left -biflatness of some hypergroup algebras in terms of amenability of their related hypergroups.

2. Approximate Biprojectivity and Left -Biprojectivity of Hypergroup Algebras

Recall that if is a Banach algebra and is a closed two-sided ideal in , then for each such that , the map is a character on .

Proposition 1. Let be a Banach algebra, , and let be a closed two-sided ideal of such that . If is left -biprojective, then is left -biprojective.

Proof. Since is left -biprojective, there is a bounded linear map such that and for every . Consider the quotient map . It may be noted that

By the assumption , we have . Thus, can be dropped on . Define by . It may be observed that

Moreover,and also

Hence, is left -biprojective.

Corollary 1. Let be a Banach algebra with a left approximate identity and . If is a closed two-sided ideal of and is left -biprojective, then is left -biprojective.

In the group case, it is well-known that the group algebra is biprojective if and only if is compact (see, for example, [4]). In the similar way, in Theorem 3.1 in [6], it is shown that if the hypergroup algebra is biprojective, then is compact. In the following, we give a generalization of this result and characterize approximate biprojectivity of the hypergroup algebra .

Theorem 1. Let be a locally compact hypergroup. If the hypergroup algebra is approximately biprojective, then is compact.

Proof. Let be approximately biprojective. Since has a bounded approximate identity (Theorem 1.6.15 in [3]), the hypergroup algebra is left -contractible for every (Theorem 3.9 in [20]). Consider the augmentation character on via

So, there exists satisfying and for every . Choose such that . It is worth noting that

Since , we have . Thus, . It follows that is a constant function. Therefore, . The latter implies that is compact (page 40 in [3]).

We study left -biprojectivity of abstract Segal algebras. So, we start with definition of abstract Segal algebras.

Definition 1. Let be a Banach algebra with norm . We say that a Banach algebra with norm is an abstract Segal algebra with respect to if(i) is a dense left ideal in A(ii)There exists such that for every (iii)There exists such that for every and Moreover, we say that is a symmetric abstract Segal algebras if is a two-sided ideal of and there exists such that for every and .
For more details about abstract Segal algebras, see [16]. We note that there are some abstract Segal algebras which are not symmetric. Homological and cohomological properties of abstract Segal algebras have been studied in many papers (see, for example, [17–19]). Recall that it is known that (see Lemma 2.2 in [1]). It is worthwhile to mention that Essmaili et al. in [6] studied right -biprojectivity (they called it condition ) of Banach algebras associated with a hypergroup, especially symmetric Segal algebras. As a main result, they showed that if is a commutative hypergroup and is a symmetric abstract Segal algebra with respect to , then is right -biprojective if and only if is compact, where . In the following, using Theorem 1, we extend the left version of (Corollary 3.11 in [6]).

Theorem 2. Let be a locally compact hypergroup, , and let be an abstract Segal algebra with respect to which possess a left approximate identity. Then, the following statements are equivalent:(i) is left -biprojective(ii) is compact

Proof. Suppose that is left -biprojective. Applying Proposition 1 in [18], is left -contractible. By Proposition 2.5 in [1], is left -contractible. By similar argument as in the proof of Theorem 1, is compact.

Conversely, let be compact with the normalized Haar measure (page 40 in [3]). Then, for each , we havefor some . So, . Put . We claim that and for every . To see this,and also

So, is left -contractible. Using Proposition 2.5 in [1] again, is left -contractible. Hence, is left -biprojective (Lemma 1 in [18]).

3. Approximate Biflatness and Left -Biflatness of Hypergroup Algebras

In this section, we study approximate biflatness and left -biflatness of some algebras related to a hypergroup.

We remind that a Banach algebra is called approximately biflat if there exists a net of -bimodule morphisms from into such that . Here, stands for the weak operator topology (see [20]).

Following [21], a locally compact hypergroup is called left amenable if there exists a left invariant mean on . That is, a bounded linear functional such that and for all and .

Proposition 2. Let be a locally compact hypergroup and let be an abstract Segal algebra with respect to which possess a left approximate identity. If is approximately biflat, then is left amenable.

Proof. Suppose that is approximately biflat and fix a character . Since has a left approximate identity, by Proposition 2. 4 in [22], is left -amenable. Now, Proposition 2.3 in [1] follows that is left -amenable. Applying Theorem 3.5 in [14], we conclude that is left amenable.

Corollary 2. Let be a locally compact hypergroup and let be an abstract Segal algebra with respect to which possess a left approximate identity. If is approximately biflat, then is left amenable.

Proof. Fix a character . Since has a left approximate identity, similar to the arguments as in the proof of (Theorem 2.2 in [23]), approximately biflatness of follows that is left -amenable. It deduces that is left -amenable. Thus, by Theorem 3.5 in [14], is left amenable.

Let be a Banach algebra and . Then, is called left -biflat if there exists a bounded linear map such that

Here, is a unique extension of which is defined by for all . The right case can be defined similarly. For further information, see [24].

Theorem 3. Let be a locally compact hypergroup, , and let be an abstract Segal algebra with respect to which possess a left approximate identity. The following statements are equivalent:(i) is left -biflat(ii) is left amenable(iii) is left -biflat

Proof. (i) (ii) Let be left -biflat. Since has a left approximate identity, by Lemma 2.1 in [21], is left -amenable. It follows that is left -amenable. Now, applying Theorem 3.5 in [14], we conclude that is left amenable.

(ii) (iii) Let be a left amenable hypergroup. Then, by Theorem 3.5 in [14], is left -amenable. Now, by Proposition 2.3 in [1], is left -amenable. Using Proposition 3.4 in [13], is left -amenable and then Lemma 2.3 in [25] implies that is left -biflat.

(iii) (i) Suppose that is left -biflat. Then, by Theorem 2.2 in [21], is left -biflat.

Suppose that and is a locally compact hypergroup. Set with the norm

becomes an abstract Segal algebra with respect to . It is known that possess a left approximate identity (see [2]).

The following corollary is an easy consequence of Proposition 2 and Corollary 2.

Corollary 3. Let be a locally compact hypergroup and . If (or ) is approximately biflat, then is left amenable.

Example 1. In this example, we give a Banach algebra which is left -biflat but it is not right -biflat.
Let be a Banach space with and be a nonzero functional such that . Define a multiplication in via , for all . It is easy to see that is a Banach algebra and . We denote the unitization of with . It is known that is a closed ideal in . Moreover, has an extension to , that is, which is given by for all and . We claim that is left -biflat but it is not right -biflat. To see this, we know that is left -amenable, applying Lemma 3.2 in [13], it follows that is left -amenable. It gives that is left -biflat. Now, assume conversely that is right -biflat. Since is unital, has an element such that and , for all . Following the similar arguments as in Theorem 1.4 in [13], we have a bounded net in such that and , for all . Pick an element in such that . Replace with , we may suppose that is a bounded net in such that and , for all . However,for all . Let be any element in and put it at above fact. It follows that . So, is an isomorphism. Therefore, which is a contradiction.

Data Availability

The data that support the findings of this study are available from all the authors.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The first author is thankful to Ilam University, for their support.

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Copyright © 2022 Amir Sahami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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